Boccardo, Lucio; Gallouët, Thierry; Vazquez, Juan Luis Solutions of nonlinear parabolic equations without growth restrictions on the data. (English) Zbl 0993.35055 Electron. J. Differ. Equ. 2001, Paper No. 60, 20 p. (2001). The paper deals with solutions in \(\mathbb{R}^N\) to nonlinear parabolic equations of the form \(u_t+L(u)+h(x,t,u)=0\), with initial condition \(u(x,0)=u_0(x)\). \(L(u)\) is an elliptic differential operator in divergence form with some structure conditions, which include the standard \(p\)-Laplacian operator, and \(h(x,t,u)\) is a function which grows uniformly with \(u\) at a sufficient rate. If \(L(u)\) is the \(p\)-Laplacian then the rate of growth of \(h\) is greather than \(p-1\). The function \(u_0(x)\) is assumed to be locally integrable in \(\mathbb{R}^N\), without any control of its growth at infinite. First, the authors establish a priori estimates of local type for a sequence of suitable approximate problems, then, passing to the limit and using the conditions in above, they prove existence of a solution for the initial problem. The corresponding results for elliptic problems were already studied. Also the regularity of the solution is investigated. In particular, for large rates of growth of \(h\) with respect to \(u\), the regularity of \(u\) and \(Du\) is improved with respect to the results known from the standard theory. Finally, the question of uniqueness of local solutions is discussed. Reviewer: G.Porru (Cagliari) Cited in 17 Documents MSC: 35K65 Degenerate parabolic equations 35K15 Initial value problems for second-order parabolic equations 35K55 Nonlinear parabolic equations Keywords:\(p\)-Laplacian operator; global existence; growth conditions; uniqueness of local solutions PDFBibTeX XMLCite \textit{L. Boccardo} et al., Electron. J. Differ. Equ. 2001, Paper No. 60, 20 p. (2001; Zbl 0993.35055) Full Text: EuDML EMIS